AdS/CFT for 3D Higher-Spin Gravity Coupled to Matter Fields

New holographic prescription for the model of 3d higher-spin gravity coupled to real matter fields $B_{\mu\nu}$ and $C$, which was introduced in ArXiv:1304.7941[hep-th], is formulated. By using a local symmetry, two of the components of $B_{\mu\nu}$ are eliminated, and gauge-fixing conditions are imposed such that the non-vanishing component, $B_{\phi\rho}$, satisfies a covariantly-constancy condition in the background of Chern-Simons gauge fields $A_{\mu}$, $\bar{A}_{\mu}$. In this model, solutions to the classical equations of motion for $A_{\mu}$ and $\bar{A}_{\mu}$ are non-flat due to the interactions with matter fields. The solutions for the gauge fields can, however, be split into two parts, flat gauge fields ${\cal A}_{\mu}$, $\bar{{\cal A}}_{\mu}$, and those terms that depend on the matter fields. The equations for the matter fields then coincide with covariantly-constancy equations in the flat backgrounds ${\cal A}_{\mu}$ and $\bar{{\cal A}}_{\mu}$, which are exactly the same as those in linearized 3d Vasiliev gravity. The two- and three-point correlation functions of operators in the boundary CFT are computed by using an on-shell action, $\text{tr}\, (B_{\phi\rho} \, C)$. This term does not depend on coordinates due to the matter equations of motion, and it is not necessary to take the near-boundary limit $\rho \rightarrow \infty$. Analysis is presented for SL(3,R) $\times$ SL(3,R) as well as $HS[\frac{1}{2}] \times HS[\frac{1}{2}]$ higher-spin gravity. In the latter model, scalar operators with scaling dimensions $\Delta_+=\frac{3}{2}$ and $\Delta_-=\frac{1}{2}$ appear in a single quantization.